Galois geometry

Number of subspaces of a Galois geometry

The Galois geometry PG(𝑛,𝑞) contains (𝑛−1𝑑−1)𝑞 subspaces of dimension 𝑑, fin where

(𝑛𝑑)𝑞=𝑑−1∏𝑖=0𝑞𝑛−𝑖−1𝑞𝑖+1−1

is the Gaußian binomial coëfficient.¹

Proof

We define

Θ𝑞(𝑟)=𝑞𝑟+1−1𝑞−1[𝑟,𝑠]𝑞=∏(𝑞𝑖−1)

Let 𝑉 =GF(𝑞)𝑛+1, so PG(𝑛,𝑞) =P(𝑉). As a vector space 𝑉 contains 𝑞𝑛 +1 points. Now the number of points in PG(𝑛,𝑞) equals the number of 1-dimensional subspaces of 𝑉, and different subspaces have only the origin in common. Thus the number of 1-dimensional subspaces is

𝑞𝑛+1−1𝑛−1=Θ𝑞(𝑛)

Each 𝑑-dimensional subspace is determined by 𝑑 +1 linearly independent points, so the number of 𝑑-dimensional subspaces in an 𝑛-dimensional projective space is the total number of independent sets of points of size 𝑑 +1 divided by the number of independent sets of points of size 𝑑 +1 in each 𝑑-dimensional subspace. This gives

[𝑛−𝑑+1,𝑛+1]𝑞[1,𝑑+1]𝑞

as required.

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Footnotes

1. 2020. Finite geometries, ¶4.7, p. 79 ↩